Without contradicting any of the excellent answers here, I have one rule of thumb which is often (but not always) decisive. (A passing comment in the answer by @Dante seems pertinent too.)

It sometimes seems too obvious to state, but here you are.

I am happy to call a distribution non-normal if I think I can offer a different description that is clearly more appropriate.

So, if there is minor curvature and/or irregularity in the tails of a normal quantile-quantile plot, but approximate straightness on a gamma quantile-quantile plot, I can say "That's not well characterised as a normal; it's more like a gamma".

It's no accident that this echoes a standard argument in history and philosophy of science, not to mention general scientific practice, that a hypothesis is most clearly and effectively refuted when you have a better one to put in its place. (Cue: allusions to Karl Popper, Thomas S. Kuhn, and so forth.)

It is true that for beginners, and indeed for everyone, there is a smooth gradation between "That is normal, except for minor irregularities which we always expect" and "That is very different from normal, except for some rough similarity which we often get".

Confidence(-like) envelopes and multiple simulated samples can help mightily, and I use and recommend both, but this can be helpful too. (Incidentally, comparing with a portfolio of simulations is a repeated recent re-invention, but goes back at least as far in Shewhart in 1931.)

I'll echo my top line. Sometimes no brand-name distribution appears to fit at all, and you have to move forward as best you can.

Without contradicting any of the excellent answers here, I have one rule of thumb which is often (but not always) decisive. (A passing comment in the answer by @Dante seems pertinent too.)

It sometimes seems too obvious to state, but here you are.

I am happy to call a distribution non-normal if I think I can offer a different description that is clearly more appropriate.

So, if there is minor curvature and/or irregularity in the tails of a normal quantile-quantile plot, but approximate straightness on a gamma quantile-quantile plot, I can say "That's not well characterised as a normal; it's more like a gamma".

It's no accident that this echoes a standard argument in history and philosophy of science, not to mention general scientific practice, that a hypothesis is most clearly and effectively refuted when you have a better one to put in its place. (Cue: allusions to Karl Popper, Thomas S. Kuhn, and so forth.)

It is true that for beginners, and indeed for everyone, there is a smooth gradation between "That is normal, except for minor irregularities which we always expect" and "That is very different from normal, except for some rough similarity which we often get".

Confidence(-like) envelopes and multiple simulated samples can help mightily, and I use and recommend both, but this can be helpful too. (Incidentally, comparing with a portfolio of simulations is a repeated recent re-invention, but goes back at least as far as Shewhart in 1931.)

I'll echo my top line. Sometimes no brand-name distribution appears to fit at all, and you have to move forward as best you can.